Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

Abstract

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive in 𝑋. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

1. Introduction

Let 𝑋 be a real Banach space whose dual space is denoted by 𝑋^∗. Let 𝑈={𝑥∈𝑋∶‖𝑥‖=1} denote the unite sphere of 𝑋. A Banach space 𝑋 is said to be uniformly convex if, for each 𝜖∈(0,2], there exists 𝛿>0 such that for all 𝑥,𝑦∈𝑈, 𝑥−𝑦≥𝜖⟹𝑥+𝑦2≤1−𝛿. (1) It is known that a uniformly convex Banach space is reflexive and strict convex. The normalized duality mapping 𝐽∶𝑋→2^𝑋∗ is defined by 𝐽(𝑥)=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖²=𝑥^∗2,∀𝑥∈𝑋, (2) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that 𝐽(𝑥) is nonempty for each 𝑥∈𝑋. Moreover, it is known that 𝐽 is single-valued if and only if 𝑋 is smooth, whereas if 𝑋 is uniformly smooth, then 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋. If 𝑋 has a uniformly Gateaux differentiable norm, then the duality mapping 𝐽 is norm-to-weak* uniformly continuous on bounded subsets of 𝑋; see for example [1].

Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝑋. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (3) The set of fixed points of 𝑇 is denoted by Fix(𝑇). We use the notation ⇀to indicate the weak convergence and → to indicate the strong convergence.

Recall that (possibly multivalued) operator 𝐴⊂𝐸×𝐸 with domain 𝐷(𝐴) and range 𝑅(𝐴) in 𝐸 is accretive if, for each 𝑥_𝑖∈𝐷(𝐴) and 𝑦_𝑖∈𝐴𝑥_𝑖 (𝑖=1,2), there exists a 𝑗(𝑥₁−𝑥₂)∈𝐽(𝑥₁−𝑥₂) such that ⟨𝑦₁−𝑦₂,(𝑥₁−𝑥₂)⟩≥0 (here 𝐽 is the duality mapping). An accretive operator 𝐴 is said to satisfy the range condition if 𝐷(𝐴)⊂𝑅(𝐼+𝑟𝐴) for all 𝑟>0. An accretive operator 𝐴 is 𝑚-accretive if 𝑅(𝐼+𝑟𝐴)=𝐸 for each 𝑟>0. If 𝐴 is an accretive operator which satisfies the range condition, then we can define, for each 𝑟>0, a mapping 𝐽_𝑟∶𝑅(𝐼+𝑟𝐴)→𝐷(𝐴) defined by 𝐽_𝑟=(𝐼+𝑟𝐴)⁻¹, which is called the resolvent of 𝐴. We know that 𝐽_𝑟 is nonexpansive and Fix(𝐽_𝑟)=𝐴⁻¹0 for all 𝑟>0. Hence Fix𝐽_𝑟=𝐴⁻¹0={𝑧∈𝐷(𝐴)∶0∈𝐴𝑧}. (4) If 𝐴⁻¹0≠∅, then the inclusion 0∈𝐴𝑧 is solvable.

The following resolvent identity is well known to us; see [2], where more details on accretive operators can be found.

Proposition 1 (resolvent identity). For 𝜆>0,𝜇>0, and 𝑥∈𝑋, 𝐽_𝜆𝑥=𝐽_𝜇𝜇𝜆𝜇𝑥+1−𝜆𝐽_𝜆𝑥. (5)

Recently, Aoyama et al. [3] studied the following iterative scheme in a uniformly convex Banach space having a uniformly Gareaux differentiable norm: for resolvents 𝐽_𝑟𝑛 of an accretive operator 𝐴 such that 𝐴⁻¹0≠∅ and 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴) and {𝛼_𝑛}⊂(0,1), 𝑥₀𝑥=𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛. (6) They proved that the sequence {𝑥_𝑛} generated by (6) converges strongly to a zero of 𝐴 under appropriate assumptions on {𝛼_𝑛} and {𝑟_𝑛}. Subsequently, Ceng et al. [4] introduced and analyzed the following composite iterative scheme in either a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping, 𝑥₀𝑦=𝑥∈𝐸,_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝐽_𝑟𝑛𝑦_𝑛, (7) where 𝑢∈𝐷(𝐴) is an arbitrary (but fixed) element, under the following control conditions:(H1)lim_𝑛→∞𝛼_𝑛=0;(H2)∑^∞_𝑛=0𝛼_𝑛=∞, or, equivalently, ∏^∞_𝑛=0(1−𝛼_𝑛)=0;(H3)∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛−1|<∞;(H4)𝑟_𝑛≥𝜀, for all 𝑛≥0, for some 𝜀>0 and ∑^∞_𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞;(H5)𝛽_𝑛∈[0,𝑎) for some 𝑎∈(0,1) and ∑^∞_𝑛=1|𝛽_𝑛−𝛽_𝑛−1|<∞.

Furthermore, as the viscosity approximation method, Jung [5] purposed and analyzed the following composite iterative scheme for finding a zero of an accretive operator 𝐴: for resolvent 𝐽_𝑟𝑛 of an accretive operator 𝐴 such that 𝐴⁻¹0≠∅ and 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴), 𝑓∈Ξ_𝐶 (Ξ_𝐶 denotes the set of all contractions on 𝐶) and {𝛼_𝑛},{𝛽_𝑛}⊂(0,1), 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝐽_𝑟𝑛𝑦_𝑛. (8) He established the strong convergence of the sequence {𝑥_𝑛} generated by (8) to a zero of 𝐴 under certain appropriate conditions.

Theorem 2 (see [5, Theorem 3.1]). Let 𝑋 be a strictly convex and reflexive Banach space having a uniformly Gateaux differentiable norm. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝐴⊂𝑋×𝑋 an accretive operator in 𝑋 such that 𝐴⁻¹0≠∅ and 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴). Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,1) which satisfy the following conditions:(C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(B1)𝛽_𝑛∈[0,𝑎) for some 0<𝑎<1 for all 𝑛≥0. Let 𝑓∈Ξ_𝐶 and 𝑥₀∈𝐶 be chosen arbitrarily. Let {𝑥_𝑛} be a sequence generated by (8) for 𝑟_𝑛>0. If {𝑥_𝑛} is asymptotically regular, that is, lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0, then {𝑥_𝑛} converges strongly to 𝑞∈𝐴⁻¹0, which is the unique solution of the variational inequality problem (VIP) 𝐼−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑓∈Ξ_𝐶,𝑝∈𝐴⁻¹0. (9)

On the other hand, we first recall the following concepts.

Definition 3. Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝑋 and let 𝐴∶𝐶→𝑋 be a mapping of 𝐶 into 𝑋. Then 𝐴 is said to be:
(i) accretive if, for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥0, (10) where 𝐽 is the normalized duality mapping;
(ii) 𝛼-strongly accretive if, for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝛼𝑥−𝑦², (11) for some 𝛼∈(0,1);
(iii) 𝛽-inverse-strongly-accretive if, for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝛽𝐴𝑥−𝐴𝑦², (12) for some 𝛽>0;
(iv) 𝜆-strictly pseudocontractive [6] if, for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜆𝑥−𝑦−𝐴𝑥−𝐴𝑦² (13) for some 𝜆∈(0,1).

It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for example, [7–9].

Very recently, Cai and Bu [10] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space 𝑋, which involves finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝐽𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝐽𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (14)where 𝐶 is a nonempty, closed, and convex subset of 𝑋, 𝐵₁,𝐵₂∶𝐶→𝑋 are two nonlinear mappings, and 𝜇₁ and 𝜇₂ are two positive constants. Here the set of solutions of GSVI (14) is denoted by GSVI(𝐶,𝐵₁,𝐵₂). In particular, if 𝑋=𝐻, a real Hilbert space, then GSVI (14) reduces to the following GSVI of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (15) where 𝜇₁ and 𝜇₂ are two positive constants. The set of solutions of problem (15) is still denoted by GSVI(𝐶,𝐵₁,𝐵₂). In particular, if 𝐵₁=𝐵₂=𝐴, then problem (15) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [11]. Furthermore, if 𝑥^∗=𝑦^∗ additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (16) The solution set of the VIP (16) is denoted by VI(𝐶,𝐴). Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [12]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous.

Recently, Ceng et al. [13] transformed problem (15) into a fixed point problem in the following way.

Lemma 4 (see [13]). For a given 𝑥,𝑦∈𝐶,(𝑥,𝑦) is a solution of problem (15) if and only if 𝑥 is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶, (17) where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥) and 𝑃_𝐶 is the projection of 𝐻 onto 𝐶.
In particular, if the mapping 𝐵_𝑖∶𝐶→𝐻 is 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2, then the mapping 𝐺 is nonexpansive provided 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2.

In 1976, Korpelevič [14] proposed an iterative algorithm for solving the VIP (16) in Euclidean space 𝐑^𝑛as follows: 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜏𝐴𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜏𝐴𝑦_𝑛,𝑛≥0, (18)with 𝜏>0 a given number, which is known as the extragradient method (see also [15]). The literature on the VIP is vast and Korpelevich's extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [10, 13, 16–23] the and references therein, to name but a few.

In particular, whenever 𝑋 is still a real smooth Banach space, 𝐵₁=𝐵₂=𝐴, and 𝑥^∗=𝑦^∗, then GSVI (17) reduces to the variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (19) which was considered by Aoyama et al. [24]. Note that VIP (19) is connected with the fixed point problem for nonlinear mapping (see, e.g., [15, 25]), the problem of finding a zero point of a nonlinear operator (see, e.g., [1, 26]), and so on. It is clear that VIP (19) extends VIP (16) from Hilbert spaces to Banach spaces.

In order to find a solution of VIP (19), Aoyama et al. [24] introduced the following iterative scheme for an accretive operator 𝐴: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛Π_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,∀𝑛≥1, (20) where Π_𝐶 is a sunny nonexpansive retraction from 𝑋 onto 𝐶. Then they proved a weak convergence theorem.

Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (14) which contains VIP (19) as a special case. Very recently, Cai and Bu [10] constructed an iterative algorithm for solving GSVI (14) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach space 𝑋.

Lemma 5 (see [27]). Let 𝑋 be a 2-uniformly smooth Banach space. Then 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽(𝑥)+2𝜅𝑦²,∀𝑥,𝑦∈𝑋, (21) where 𝜅 is the 2-uniformly smooth constant of 𝑋 and 𝐽 is the normalized duality mapping from 𝑋 into 𝑋^∗.

Define the mapping 𝐺∶𝐶→𝐶 as follows: 𝐺(𝑥)∶=Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (22) The fixed point set of 𝐺 is denoted by Ω. Then their strong convergence theorem on the proposed method is stated as follows.

Theorem 6 (see [10, Theorem 3.1]). Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛽_𝑖-inverse strongly accretive with 0<𝜇_𝑖<𝛽_𝑖/𝜅² for 𝑖=1,2. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛿∈(0,1). Let {𝑆_𝑛}^∞_𝑛=1 be an infinite family of nonexpansive mappings of 𝐶 into itself such that 𝐹=⋂^∞_𝑖=1Fix(𝑆_𝑖)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺 defined by (22). For arbitrarily given 𝑥₁∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆_𝑛𝑦_𝑛,𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛,𝑧_𝑛=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛,𝑢_𝑛=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,∀𝑛≥1. (23)Suppose that {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) satisfying the following conditions:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Assume that ∑^∞_𝑛=1sup_𝑥∈𝐷‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖<∞ for any bounded subset 𝐷 of 𝐶 and let 𝑆 be a mapping of 𝐶 into 𝑋 defined by 𝑆𝑥=lim_𝑛→∞𝑆_𝑛𝑥 for all 𝑥∈𝐶 and suppose that Fix(𝑆)=⋂^∞_𝑛=1Fix(𝑆_𝑛). Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (24)

Corollary 7 (see [10, Corollary 3.2]). Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛽_𝑖-inverse strongly accretive with 0<𝜇_𝑖<𝛽_𝑖/𝜅² for 𝑖=1,2. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛿∈(0,1). Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that 𝐹=Fix(𝑆)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺 defined by (22). For arbitrarily given 𝑥₁∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑦_𝑛,𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛,𝑧_𝑛=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛,𝑢_𝑛=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,∀𝑛≥1. (25)Suppose that {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) satisfying the following conditions:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (26)

We remark that in Theorem 6, the Banach space 𝑋 is both uniformly convex and 2-uniformly smooth. According to Lemma 5, the 2-uniform smoothness of 𝑋 guarantees the nonexpansivity of the mapping 𝐼−𝜇_𝑖𝐵_𝑖 for 𝛼_𝑖-inverse-strongly accretive mapping 𝐵_𝑖∶𝐶→𝑋 with 0≤𝜇_𝑖≤𝛼_𝑖/𝜅² for 𝑖=1,2, and hence the composite mapping 𝐺∶𝐶→𝐶 is nonexpansive where 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂). In the meantime, for the convenience of implementing the argument techniques in [13], they have applied the following inequality in a real smooth and uniform convex Banach space 𝑋.

Proposition 8 (see [28]). Let 𝑋 be a real smooth and uniform convex Banach space and let 𝑟>0. Then there exists a strictly increasing, continuous, and convex function 𝑔∶[0,2𝑟]→𝐑, 𝑔(0)=0 such that 𝑔𝑥−𝑦≤‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐵_𝑟, (27) where 𝐵_𝑟={𝑥∈𝑋∶‖𝑥‖≤𝑟}.

Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Motivated and inspired by the research going on this area, we introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator 𝐴⊂𝑋×𝑋 such that 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴) and solving GSVI (14) and a fixed point problem of an infinite family of nonexpansive self-mappings on 𝐶. We establish some strong convergence theorems for hybrid implicit and explicit extragradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings on 𝐶 and a self-mapping whose complement is strictly pseudocontractive and strongly accretive on 𝐶. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see, for example, [5, 10, 13, 16].

2. Preliminaries

Let 𝑋 be a real Banach space. 𝑋 is said to be smooth if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡, (28) exists for all 𝑥,𝑦∈𝑈; in this case, 𝑋 is also said to have a Gateaux differentiable norm. 𝑋 is said to have a uniformly Gateaux differentiable norm if for each 𝑦∈𝑈, the limit is attained uniformly for 𝑥∈𝑈. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for 𝑥,𝑦∈𝑈. The norm of 𝑋 is said to be the Frechet differential if, for each 𝑥∈𝑈, this limit is attained uniformly for 𝑦∈𝑈. In addition, we define a function 𝜌∶[0,∞)→[0,∞) called the modulus of smoothness of 𝑋 as follows: 𝜌(𝜏)1=sup2+‖𝑥‖𝑦.𝑥+𝑦𝑥−𝑦−1∶𝑥,𝑦∈𝑋,=1,=𝜏 (29) It is known that 𝑋 is uniformly smooth if and only if lim_𝜏→0𝜌(𝜏)/𝜏=0. Let 𝑞 be a fixed real number with 1<𝑞≤2. Then a Banach space 𝑋 is said to be 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌(𝜏)≤𝑐𝜏^𝑞 for all 𝜏>0. As pointed out in [29], no Banach space is 𝑞-uniformly smooth for 𝑞>2.

We list some lemmas that will be used in the sequel. Lemma 9 can be found in [30]. Lemma 10 is an immediate consequence of the subdifferential inequality of the function (1/2)‖⋅‖².

Lemma 9. Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝛽_𝑛+𝛾_𝑛,∀𝑛≥0, (30) where {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} satisfy the following conditions:(i){𝛼_𝑛}⊂[0,1] and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii)limsup_𝑛→∞𝛽_𝑛≤0;(iii)𝛾_𝑛≥0, for all 𝑛≥0, and ∑^∞_𝑛=0𝛾_𝑛<∞.Then limsup_𝑛→∞𝑠_𝑛=0.

Lemma 10. In a smooth Banach space 𝑋, there holds the inequality 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦,∀𝑥,𝑦∈𝑋. (31)

Lemma 11 (see [6]). Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space 𝑋 and let 𝐴∶𝐶→𝑋 be a 𝜆-strictly pseudocontractive mapping. Then 𝐴 is Lipschitz continuous with constant 1+1/𝜆.

Proof. Since 𝐴∶𝐶→𝑋 is a 𝜆-strictly pseudocontractive mapping, we have for all 𝑥,𝑦∈𝐶𝐴𝑥−𝐴𝑦,𝐽𝑥−𝑦≤𝑥−𝑦²−𝜆𝑥−𝑦−𝐴𝑥−𝐴𝑦², (32) and hence 𝜆(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦²≤(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦,𝐽𝑥−𝑦. (33) This yields )()𝑦≤1(𝐼−𝐴𝑥−𝐼−𝐴𝜆𝑥−𝑦,∀𝑥,𝑦∈𝐶. (34) Therefore, ≤+)()𝑦≤1𝐴𝑥−𝐴𝑦𝑥−𝑦(𝐼−𝐴𝑥−𝐼−𝐴1+𝜆𝑥−𝑦,∀𝑥,𝑦∈𝐶. (35)

Let 𝐷 be a subset of 𝐶 and let Π be a mapping of 𝐶 into 𝐷. Then Π is said to be sunny if Π[Π(𝑥)+𝑡(𝑥−Π(𝑥))]=Π(𝑥), (36) whenever Π(𝑥)+𝑡(𝑥−Π(𝑥))∈𝐶 for 𝑥∈𝐶 and 𝑡≥0. A mapping Π of 𝐶 into itself is called a retraction if Π²=Π. If a mapping Π of 𝐶 into itself is a retraction, then Π(𝑧)=𝑧 for every 𝑧∈𝑅(Π) where 𝑅(Π) is the range of Π. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷. The following lemma concerns the sunny nonexpansive retraction.

Lemma 12 (see [31]). Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space 𝑋. Let 𝐷 be a nonempty subset of 𝐶. Let Π be a retraction of 𝐶 onto 𝐷. Then the following are equivalent:(i)Π is sunny and nonexpansive;(ii)‖Π(𝑥)−Π(𝑦)‖²≤⟨𝑥−𝑦,𝐽(Π(𝑥)−Π(𝑦))⟩, for all 𝑥,𝑦∈𝐶;(iii)⟨𝑥−Π(𝑥),𝐽(𝑦−Π(𝑥))⟩≤0, for all 𝑥∈𝐶,𝑦∈𝐷.

It is well known that if 𝑋=𝐻 a Hilbert space, then a sunny nonexpansive retraction Π_𝐶 is coincident with the metric projection from 𝑋 onto 𝐶; that is, Π_𝐶=𝑃_𝐶. If 𝐶 is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space 𝑋 and if 𝑇∶𝐶→𝐶 is a nonexpansive mapping with the fixed point set Fix(𝑇)≠∅, then the set Fix(𝑇) is a sunny nonexpansive retract of 𝐶.

Lemma 13 (see [32]). Let 𝑋 be a uniformly convex Banach space and 𝐵_𝑟={𝑥∈𝑋∶‖𝑥‖≤𝑟},𝑟>0. Then there exists a continuous, strictly increasing, and convex function 𝑔∶[0,∞]→[0,∞],𝑔(0)=0 such that 𝛼𝑥+𝛽𝑦+𝛾𝑧²≤𝛼‖𝑥‖²+𝛽𝑦²+𝛾‖𝑧‖²−𝛼𝛽𝑔𝑥−𝑦 (37)for all 𝑥,𝑦,𝑧∈𝐵_𝑟 and all 𝛼,𝛽,𝛾∈[0,1] with 𝛼+𝛽+𝛾=1.

Lemma 14 (see [33]). Let 𝐶 be a nonempty closed convex subset of a Banach space 𝑋. Let 𝑆₀,𝑆₁,… be a sequence of mappings of 𝐶 into itself. Suppose that ∑^∞_𝑛=1sup{‖𝑆_𝑛𝑥−𝑆_𝑛−1𝑥‖∶𝑥∈𝐶}<∞. Then for each 𝑦∈𝐶, {𝑆_𝑛𝑦} converges strongly to some point of 𝐶. Moreover, let 𝑆 be a mapping of 𝐶 into itself defined by 𝑆𝑦=lim_𝑛→∞𝑆_𝑛𝑦 for all 𝑦∈𝐶. Then lim_𝑛→∞sup{‖𝑆𝑥−𝑆_𝑛𝑥‖∶𝑥∈𝐶}=0.

Let 𝐶 be a nonempty closed convex subset of a Banach space 𝑋 and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. As previously mentioned, let Ξ_𝐶 be the set of all contractions on 𝐶. For 𝑡∈(0,1) and 𝑓∈Ξ_𝐶, let 𝑥_𝑡∈𝐶 be the unique fixed point of the contraction 𝑥↦𝑡𝑓(𝑥)+(1−𝑡)𝑇𝑥 on 𝐶; that is, 𝑥_𝑡=𝑡𝑓𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡. (38)

Lemma 15 (see [15]). Let 𝑋 be a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let 𝐶 a nonempty closed convex subset of 𝑋,𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅, and 𝑓∈Ξ_𝐶. Then the net {𝑥_𝑡} defined by 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑇𝑥_𝑡 converges strongly to a point in Fix(𝑇). If we define a mapping 𝑄∶Ξ_𝐶→Fix(𝑇) by 𝑄(𝑓)∶=𝑠−lim_𝑡→0𝑥_𝑡, for all 𝑓∈Ξ_𝐶, then 𝑄(𝑓) solves the following VIP: 𝐼−𝑓𝑄𝑓,𝐽𝑄𝑓−𝑝≤0,∀𝑓∈Ξ_𝐶,𝑝∈Fix(𝑇). (39)

Lemma 16 (see [34]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇_𝑛}^∞_𝑛=0 be a sequence of nonexpansive mappings on 𝐶. Suppose ⋂^∞_𝑛=0Fix(𝑇_𝑛) is nonempty. Let {𝜆_𝑛} be a sequence of positive numbers with ∑^∞_𝑛=0𝜆_𝑛=1. Then a mapping 𝑆 on 𝐶 defined by 𝑆𝑥=∑^∞_𝑛=0𝜆_𝑛𝑇_𝑛𝑥 for 𝑥∈𝐶 is well defined, nonexpansive and Fix(𝑆)=⋂^∞_𝑛=0Fix(𝑇_𝑛) holds.

Lemma 17 (see [27]). Given a number 𝑟>0. A real Banach space 𝑋 is uniformly convex if and only if there exists a continuous strictly increasing function 𝑔∶[0,∞)→[0,∞), 𝑔(0)=0, such that 𝜆𝑥+(1−𝜆)𝑦²≤𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑔𝑥−𝑦 (40)for all 𝜆∈[0,1] and 𝑥,𝑦∈𝑋 such that ‖𝑥‖≤𝑟 and ‖𝑦‖≤𝑟.

Lemma 18 (see [24]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶 and let 𝐴 be an accretive operator of 𝐶 into 𝑋. Then, for all 𝜆>0, VI(𝐶,𝐴)=FixΠ_𝐶(𝐼−𝜆𝐴). (41)

Lemma 19. Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋 and let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝜁_𝑖 strictly pseudocontractive and 𝜃_𝑖 strongly accretive with 𝜃_𝑖+𝜁_𝑖≥1 for 𝑖=1,2. Then, for 𝜇_𝑖∈(0,1], we have 𝐼−𝜇_𝑖𝐵_𝑖𝑥−𝐼−𝜇_𝑖𝐵_𝑖𝑦≤1−𝜃_𝑖𝜁_𝑖+1−𝜇_𝑖11+𝜁_𝑖×𝑥−𝑦,∀𝑥,𝑦∈𝐶, (42) for 𝑖=1,2. In particular, if 1−(𝜁_𝑖/(1+𝜁_𝑖))(1−(1−𝜃_𝑖)/𝜁_𝑖)≤𝜇_𝑖≤1, then 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2.

Proof. Taking into account the 𝜁_𝑖-strict pseudocontractivity of 𝐵_𝑖, by Lemma 11 we derive for every 𝑥,𝑦∈𝐶𝐵_𝑖𝑥−𝐵_𝑖𝑦≤11+𝜁_𝑖𝑥−𝑦. (43) Utilizing the 𝜃_𝑖-strong accretivity and 𝜁_𝑖-strict pseudocontractivity of 𝐵_𝑖, we get 𝜁_𝑖𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦²≤𝑥−𝑦²−𝐵_𝑖𝑥−𝐵_𝑖≤𝑦,𝐽𝑥−𝑦1−𝜃_𝑖𝑥−𝑦². (44) So, we have 𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦≤1−𝜃_𝑖𝜁_𝑖𝑥−𝑦. (45) Therefore, for 𝜇_𝑖∈(0,1], we have 𝐼−𝜇_𝑖𝐵_𝑖𝑥−𝐼−𝜇_𝑖𝐵_𝑖𝑦≤𝐼−𝐵_𝑖𝑥−𝐼−𝐵_𝑖𝑦+1−𝜇_𝑖𝐵_𝑖𝑥−𝐵_𝑖𝑦≤1−𝜃_𝑖𝜁_𝑖𝑥−𝑦+1−𝜇_𝑖11+𝜁_𝑖=𝑥−𝑦1−𝜃_𝑖𝜁_𝑖+1−𝜇_𝑖11+𝜁_𝑖𝑥−𝑦. (46) Since 1−(𝜁_𝑖/(1+𝜁_𝑖))(1−(1−𝜃_𝑖)/𝜁_𝑖)≤𝜇_𝑖≤1, it follows immediately that 1−𝜃_𝑖𝜁_𝑖+1−𝜇_𝑖11+𝜁_𝑖≤1. (47) This implies that 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2.

Lemma 20. Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶 and let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝜁_𝑖 strictly pseudocontractive and 𝜃_𝑖 strongly accretive with 𝜃_𝑖+𝜁_𝑖≥1 for 𝑖=1,2. Let 𝐺∶𝐶→𝐶 be the mapping defined by 𝐺(𝑥)=Π_𝐶Π_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁Π_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (48) If 1−(𝜁_𝑖/(1+𝜁_𝑖))(1−(1−𝜃_𝑖)/𝜁_𝑖)≤𝜇_𝑖≤1, then 𝐺∶𝐶→𝐶 is nonexpansive.

Proof. According to Lemma 19, we know that 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2. Hence, for all 𝑥,𝑦∈𝐶, we have 𝐺(𝑥)−𝐺𝑦=Π_𝐶Π_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁Π_𝐶𝑥−𝜇₂𝐵₂𝑥−Π_𝐶Π_𝐶𝑦−𝜇₂𝐵₂𝑦−𝜇₁𝐵₁Π_𝐶𝑦−𝜇₂𝐵₂𝑦=Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥−Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑦≤𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥−𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑦≤Π_𝐶𝐼−𝜇₂𝐵₂𝑥−Π_𝐶𝐼−𝜇₂𝐵₂𝑦≤𝐼−𝜇₂𝐵₂𝑥−𝐼−𝜇₂𝐵₂𝑦≤𝑥−𝑦. (49) This shows that 𝐺∶𝐶→𝐶 is nonexpansive. This completes the proof.

Lemma 21. Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶 and let 𝐵₁,𝐵₂∶𝐶→𝑋 be two nonlinear mappings. For a given 𝑥^∗,𝑦^∗∈𝐶,(𝑥^∗,𝑦^∗) is a solution of GSVI (14) if and only if 𝑥^∗=Π_𝐶(𝑦^∗−𝜇₁𝐵₁𝑦^∗) where 𝑦^∗=Π_𝐶(𝑥^∗−𝜇₂𝐵₂𝑥^∗).

Proof. We can rewrite GSVI (14) as 𝑥^∗−𝑦^∗−𝜇₁𝐵₁𝑦^∗,𝐽𝑥−𝑥^∗𝑦≥0,∀𝑥∈𝐶,_∗−𝑥^∗−𝜇₂𝐵₂𝑥^∗,𝐽𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (50) which is obviously equivalent to 𝑥^∗=Π_𝐶𝑦^∗−𝜇₁𝐵₁𝑦^∗,𝑦^∗=Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (51) because of Lemma 12. This completes the proof.

Remark 22. By Lemma 21, we observe that 𝑥^∗=Π_𝐶Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗−𝜇₁𝐵₁Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (52) which implies that 𝑥^∗ is a fixed point of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂). Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by Ω.

3. Hybrid Implicit Extragradient Algorithm

In this section, let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. We suggest and analyze a hybrid implicit extragradient algorithm for finding a zero of 𝐴⊂𝑋×𝑋 an accretive operator in 𝑋 with 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴) and solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive self-mappings in 𝑋.

Theorem 23. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝐴⊂𝑋×𝑋 be an accretive operator in 𝑋 such that 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴). Let 𝐵_𝑖∶𝐶→𝑋 be 𝜁_𝑖 strictly pseudocontractive and 𝜃_𝑖 strongly accretive with 𝜃_𝑖+𝜁_𝑖≥1 for each 𝑖=1,2. Define the mapping 𝐺∶𝐶→𝐶 by 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) where 1−(𝜁_𝑖/(1+𝜁_𝑖))(1−(1−𝜃_𝑖)/𝜁_𝑖)≤𝜇_𝑖≤1 for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be an infinite family of nonexpansive mappings of 𝐶 into itself such that 𝐹=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω∩𝐴⁻¹0≠∅ where Ω=Fix(𝐺). For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛Π_𝐶𝐼−𝜇₁𝐵₁×Π_𝐶𝐼−𝜇₂𝐵₂𝑦_𝑛,∀𝑛≥0. (53) Suppose that {𝑟_𝑛}⊂(0,∞), {𝜎_𝑛},{𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1], 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1 and the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)∑^∞_𝑛=1(|𝜎_𝑛−𝜎_𝑛−1|+|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)<∞;(iv)∑^∞_𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞ and 𝑟_𝑛≥𝜀>0, for all 𝑛≥0 for some 𝜀>0;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1.Assume that ∑^∞_𝑛=0sup_𝑥∈𝐷‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖<∞ for any bounded subset 𝐷 of 𝐶 and let 𝑆 be a mapping of 𝐶 into itself defined by 𝑆𝑥=lim_𝑛→∞𝑆_𝑛𝑥 for all 𝑥∈𝐶 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (54)

Proof. It is easy to see that (53) can be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛾_𝑛𝐵𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛𝐺𝑦_𝑛,∀𝑛≥0, (55) where 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂). By Lemma 20 we know that 𝐺 is a nonexpansive self-mapping on 𝐶.
Now, let us show that the sequence {𝑥_𝑛} is bounded. Indeed, take a fixed 𝑝∈𝐹 arbitrarily. Then from (55), we have 𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝑓𝑝−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝜌𝑥_𝑛−𝑝+𝑓𝑝−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑦_𝑛=−𝑝1−𝛿_𝑛−1−𝜌𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝+𝛿_𝑛𝑦_𝑛−𝑝, (56) which hence implies that 𝑦_𝑛≤𝛼−𝑝1−1−𝜌_𝑛1−𝛿_𝑛𝑥_𝑛+𝛼−𝑝_𝑛1−𝛿_𝑛𝑓𝑝−𝑝. (57) So, we have 𝑥_𝑛+1−𝑝≤𝜎_𝑛𝑦_𝑛+−𝑝1−𝜎_𝑛𝐺𝑦_𝑛−𝑝≤𝜎_𝑛𝑦_𝑛−𝑝+1−𝜎_𝑛𝑦_𝑛=𝑦−𝑝_𝑛≤𝛼−𝑝1−1−𝜌_𝑛1−𝛿_𝑛𝑥_𝑛+𝛼−𝑝_𝑛1−𝛿_𝑛𝑓𝑝=𝛼−𝑝1−1−𝜌_𝑛1−𝛿_𝑛𝑥_𝑛+𝛼−𝑝1−𝜌_𝑛1−𝛿_𝑛𝑓𝑝−𝑝𝑥1−𝜌≤max_𝑛,𝑓𝑝−𝑝−𝑝.1−𝜌 (58) By induction, we obtain 𝑥_𝑛−𝑝≤max𝑥₀,𝑓𝑝−𝑝−𝑝1−𝜌,∀𝑛≥0. (59) Hence {𝑥_𝑛} is bounded and so are {𝑦_𝑛}, {𝐺𝑦_𝑛}, {𝑓(𝑥_𝑛)}, {𝐽_𝑟𝑛𝑥_𝑛}, and {𝑆_𝑛𝑦_𝑛}.
Let us show that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (60) As a matter of fact, observe that 𝑦_𝑛 can be rewritten as follows: 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛, (61) where 𝑧_𝑛=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛)/(1−𝛽_𝑛). Observe that 𝑧_𝑛−𝑧_𝑛−1=𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛1−𝛽_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝑦_𝑛−11−𝛽_𝑛−1=𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛+𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1≤𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛+𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=11−𝛽_𝑛𝑦_𝑛−𝛽_𝑛𝑥_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛𝑦_𝑛−𝛽_𝑛𝑥_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1−𝛿_𝑛−1𝑆_𝑛−1𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑆_𝑛−1𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛×𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1. (62) On the other hand, if 𝑟_𝑛−1≤𝑟_𝑛, using the resolvent identity in Proposition 1, 𝐽_𝑟𝑛𝑥_𝑛=𝐽_𝑟𝑛−1𝑟_𝑛−1𝑟_𝑛𝑥_𝑛+𝑟1−^𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑛, (63) we get 𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1=𝐽_𝑟𝑛−1𝑟_𝑛−1𝑟_𝑛𝑥_𝑛+𝑟1−^𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1≤𝑟_𝑛−1𝑟_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑟1−^𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+1𝜀𝑟_𝑛−𝑟_𝑛−1𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1. (64) If 𝑟_𝑛≤𝑟_𝑛−1, we derive in a similar way 𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+1𝜀𝑟_𝑛−1−𝑟_𝑛𝐽_𝑟𝑛−1𝑥_𝑛−1−𝑥_𝑛. (65) Thus, combining the above cases we obtain 𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+𝑀₀𝑟_𝑛−1−𝑟_𝑛,∀𝑛≥1, (66) where sup_𝑛≥1{(1/𝜀)(‖𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1‖+‖𝐽_𝑟𝑛−1𝑥_𝑛−1−𝑥_𝑛‖)}≤𝑀₀ for some 𝑀₀>0. Substituting (66) into (62), we have 𝑧_𝑛−𝑧_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛾_𝑛𝑥_𝑛−1−𝑥_𝑛+𝑀₀𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑆_𝑛−1𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛𝑥_𝑛−1−𝑥_𝑛+𝑀₀𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑆_𝑛𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝑦_𝑛−1+𝛿_𝑛𝑆_𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛×𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝑥_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝜌+𝛾_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛿_𝑛𝑦_𝑛−𝑦_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛𝛽_𝑛−𝛽_𝑛−1𝑀, (67) where sup_𝑛≥0{𝑀₀+‖𝑓(𝑥_𝑛)‖+‖𝐽_𝑟𝑛𝑥_𝑛‖+‖𝑆_𝑛𝑦_𝑛‖}≤𝑀 for some 𝑀>0. In the meantime, simple calculations show that 𝑦_𝑛−𝑦_𝑛−1=𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛𝑧_𝑛−𝑧_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1−𝑧_𝑛−1. (68) Taking into account condition (iv), we may assume, without loss of generality, that {𝛽_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1). Hence, from (67) and (68) we deduce that 𝑦_𝑛−𝑦_𝑛−1≤𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛𝑧_𝑛−𝑧_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1−𝑧_𝑛−1≤𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛×11−𝛽_𝑛𝛼_𝑛𝜌+𝛾_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛿_𝑛𝑦_𝑛−𝑦_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛𝛽_𝑛−𝛽_𝑛−1𝑀+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1−𝑧_𝑛−1=𝛼_𝑛𝜌+𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛿_𝑛𝑦_𝑛−𝑦_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1+11−𝛽_𝑛−1𝛽_𝑛−𝛽_𝑛−1𝑀+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1−𝑧_𝑛−1=1−𝛿_𝑛−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝛿_𝑛𝑦_𝑛−𝑦_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1+11−𝛽_𝑛−1𝛽_𝑛−𝛽_𝑛−1𝑀+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1−𝑧_𝑛−1≤1−𝛿_𝑛−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥_𝑛−1+𝛿_𝑛𝑦_𝑛−𝑦_𝑛−1+𝑀₀𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1, (69) where sup_𝑛≥0{(𝑀/(1−𝑏))+‖𝑥_𝑛−𝑧_𝑛‖}≤𝑀₀ for some 𝑀₀>0. Thus, from condition (ii) we immediately get 𝑦_𝑛−𝑦_𝑛−1≤1−1−𝜌1−𝛿_𝑛𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₀1−𝛿_𝑛𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛1−𝛿_𝑛𝑆_𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1≤1−1−𝜌1−𝛿_𝑛𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₀𝛼1−𝑑^𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝑑𝑆1−𝑑^𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1. (70) Also, from (55) we have 𝑥_𝑛+1−𝑥_𝑛=𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1+1−𝜎_𝑛𝐺𝑦_𝑛−𝐺𝑦_𝑛−1. (71) Taking into account condition (v), we may assume, without loss of generality, that {𝛽_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1). This together with (70) implies that 𝑥_𝑛+1−𝑥_𝑛≤𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1+1−𝜎_𝑛𝐺𝑦_𝑛−𝐺𝑦_𝑛−1≤𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1+1−𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1=𝑦_𝑛−𝑦_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1≤1−1−𝜌1−𝛿_𝑛𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₀𝛼1−𝑑^𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝑑𝑆1−𝑑^𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1≤1−1−𝜌1−𝛿_𝑛𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝜎_𝑛−𝜎_𝑛−1+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝑑𝑆1−𝑑^𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1, (72) where sup_𝑛≥0{(𝑀₀/(1−𝑑))+‖𝑦_𝑛−𝐺𝑦_𝑛‖}≤𝑀₁ for some 𝑀₁>0. Since ∑^∞_𝑛=0𝛼_𝑛=∞ and ((1−𝜌)/(1−𝛿_𝑛))𝛼_𝑛≥(1−𝜌)𝛼_𝑛, we know that ∑^∞_𝑛=0((1−𝜌)/(1−𝛿_𝑛))𝛼_𝑛=∞. So, applying Lemma 9 to (72), we obtain from conditions (iii), (iv), and the assumption on {𝑆_𝑛} that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (73)
Next we show that ‖𝑥_𝑛−𝐺𝑥_𝑛‖→0 as 𝑛→∞.
Indeed, according to Lemma 10 we have from (55) 𝑦_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝≤𝛼_𝑛𝜌𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛=𝛼−𝑝_𝑛𝜌+𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛≤−𝑝1−𝛿_𝑛−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝, (74) which hence implies that 𝑦_𝑛−𝑝²≤𝛼1−^𝑛1−𝜌1−𝛿_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛1−𝛿_𝑛2𝑓𝑝−𝑝,𝐽𝑦_𝑛≤𝛼−𝑝1−^𝑛1−𝜌1−𝛿_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛1−𝛿_𝑛2𝑓𝑝−𝑝𝑦_𝑛−𝑝. (75) Utilizing Lemma 17, we get from (55) and (75) 𝑥_𝑛+1−𝑝²=𝜎_𝑛𝑦_𝑛+−𝑝1−𝜎_𝑛𝐺𝑦_𝑛−𝑝²≤𝜎_𝑛𝑦_𝑛−𝑝²+1−𝜎_𝑛𝐺𝑦_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛≤𝜎_𝑛𝑦_𝑛−𝑝²+1−𝜎_𝑛𝑦_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛=𝑦_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛≤𝛼1−^𝑛1−𝜌1−𝛿_𝑛𝑥_𝑛−𝑝²+𝛼_𝑛1−𝛿_𝑛2𝑓𝑝−𝑝𝑦_𝑛−𝑝−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛≤𝑥_𝑛−𝑝²+𝛼_𝑛1−𝛿_𝑛2𝑓𝑝−𝑝𝑦_𝑛−𝑝−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛, (76) which hence yields 𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛼_𝑛1−𝛿_𝑛2𝑓𝑝−𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1×𝑥−𝑝_𝑛−𝑥_𝑛+1+𝛼_𝑛2𝑓𝑝𝑦1−𝑑−𝑝_𝑛−𝑝. (77) Since 𝛼_𝑛→0 and ‖𝑥_𝑛+1−𝑥_𝑛‖→0, from condition (iv) and the boundedness of {𝑥_𝑛} and {𝑦_𝑛}, it follows that lim_𝑛→∞𝑔₁𝑦_𝑛−𝐺𝑦_𝑛=0. (78) Utilizing the properties of 𝑔₁, we have lim_𝑛→∞𝑦_𝑛−𝐺𝑦_𝑛=0. (79) Observe that 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑦_𝑛=𝑥_𝑛−𝑥_𝑛+1+1−𝜎_𝑛𝐺𝑦_𝑛−𝑦_𝑛⟶0,as𝑛⟶∞, (80) and hence 𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐺𝑦_𝑛+𝐺𝑦_𝑛−𝐺𝑥_𝑛≤2𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐺𝑦_𝑛⟶0,as𝑛⟶∞. (81)
That is, lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (82) Next, let us show that lim_𝑛→∞𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝑆_𝑛𝑥_𝑛−𝑥_𝑛=0. (83) Indeed, observe that 𝑦_𝑛 can be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛𝛾_𝑛+𝛿_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝑒_𝑛𝑧_𝑛, (84) where 𝑒_𝑛=𝛾_𝑛+𝛿_𝑛 and 𝑧_𝑛=(𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛)/(𝛾_𝑛+𝛿_𝑛). Utilizing Lemma 13 and (84), we have 𝑦_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝑒_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝑒_𝑛𝑧_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿−𝑝_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛+𝛿−𝑝_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛼_𝑛−𝛿_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛿_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛, (85) which hence implies that 𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛿_𝑛𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛿_𝑛𝑥_𝑛+𝑦−𝑝_𝑛−𝑝𝑥_𝑛−𝑦_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (86) Utilizing (82), conditions (i), (ii), and (v), and the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, and {𝑓(𝑥_𝑛)}, we get lim_𝑛→∞𝑔₂𝑧_𝑛−𝑥_𝑛=0. (87) From the properties of 𝑔₂, we have lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (88) Utilizing Lemma 17 and the definition of 𝑧_𝑛, we have 𝑧_𝑛−𝑝²=𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿−𝑝_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝²≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₃𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₃𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛, (89) which leads to 𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₃𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²−𝑧_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝²−𝑧_𝑛−𝑝²≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝+𝑧_𝑛−𝑝𝑥_𝑛−𝑧_𝑛+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝+𝑧_𝑛−𝑝𝑦_𝑛−𝑧_𝑛≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝+𝑧_𝑛−𝑝𝑥_𝑛−𝑧_𝑛+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑦_𝑛−𝑝+𝑧_𝑛×𝑦−𝑝_𝑛−𝑥_𝑛+𝑥_𝑛−𝑧_𝑛. (90) Since {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} are bounded, ‖𝑥_𝑛−𝑦_𝑛‖→0, and ‖𝑧_𝑛−𝑥_𝑛‖→0 as 𝑛→∞, we deduce from condition (ii) that lim_𝑛→∞𝑔₃𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛=0. (91) From the properties of 𝑔₃, we have lim_𝑛→∞𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛=0. (92) On the other hand, 𝑦_𝑛 can also be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛼_𝑛+𝛿_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛𝛼_𝑛+𝛿_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝑑_𝑛𝑧_𝑛, (93) where 𝑑_𝑛=𝛼_𝑛+𝛿_𝑛 and 𝑧_𝑛=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛿_𝑛𝑆_𝑛𝑦_𝑛)/(𝛼_𝑛+𝛿_𝑛). Utilizing Lemma 13 and the convexity of ‖⋅‖², we have 𝑦_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝+𝑑_𝑛𝑧_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝑧_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛𝛼_𝑛+𝛿_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛+𝛿−𝑝_𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛−𝑝²+𝛿_𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛼_𝑛−𝛿_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛿_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑦_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛, (94) which hence implies that 𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛿_𝑛𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+1−𝛿_𝑛𝑥_𝑛−𝑝+𝑦_𝑛×𝑥−𝑝_𝑛−𝑦_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (95) From (82), conditions (i), (ii), and (v), and the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, and {𝑓(𝑥_𝑛)}, we have lim_𝑛→∞𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=0. (96) Utilizing the properties of 𝑔₄, we have lim_𝑛→∞𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=0. (97) Note that 𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛+𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛+𝑆_𝑛𝑦_𝑛−𝑆_𝑛𝑥_𝑛≤𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛+𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑦_𝑛+𝑦_𝑛−𝑥_𝑛. (98) From (82), (92), and (97), we get lim_𝑛→∞𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (99) In terms of (99) and Lemma 14, we have 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛⟶0,as𝑛⟶∞. (100) That is, lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (101) Furthermore, we claim that lim_𝑛→∞‖𝑥_𝑛−𝐽_𝑟𝑥_𝑛‖=0 for a fixed number 𝑟 such that 𝜀>𝑟>0. In fact, from the resolvent identity in Proposition 1, we have 𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑥_𝑛=𝐽_𝑟𝑟𝑟_𝑛𝑥_𝑛+𝑟1−𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑥_𝑛≤𝑟1−𝑟_𝑛𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛. (102) So, we get from (102) 𝑥_𝑛−𝐽_𝑟𝑥_𝑛≤𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛+𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑥_𝑛≤𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛+𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=2𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛. (103) Thus, from (97) it follows that lim_𝑛→∞𝑥_𝑛−𝐽_𝑟𝑥_𝑛=0. (104) Define a mapping 𝑊𝑥=(1−𝜃₁−𝜃₂)𝐽_𝑟𝑥+𝜃₁𝑆𝑥+𝜃₂𝐺𝑥, where 𝜃₁,𝜃₂∈(0,1) are two constants with 𝜃₁+𝜃₂<1. Then, by Lemma 16, we have that Fix(𝑊)=Fix(𝐽_𝑟)∩Fix(𝑆)∩Fix(𝐺)=𝐹. We observe that 𝑥_𝑛−𝑊𝑥_𝑛=1−𝜃₁−𝜃₂𝑥_𝑛−𝐽_𝑟𝑥_𝑛+𝜃₁𝑥_𝑛−𝑆𝑥_𝑛+𝜃₂𝑥_𝑛−𝐺𝑥_𝑛≤1−𝜃₁−𝜃₂𝑥_𝑛−𝐽_𝑟𝑥_𝑛+𝜃₁𝑥_𝑛−𝑆𝑥_𝑛+𝜃₂𝑥_𝑛−𝐺𝑥_𝑛. (105) From (82), (101), and (104), we obtain lim_𝑛→∞𝑥_𝑛−𝑊𝑥_𝑛=0. (106)
Now, we claim that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0, (107) where 𝑞=𝑠−lim_𝑡→0𝑥_𝑡 with 𝑥_𝑡 being the fixed point of the contraction 𝑥↦𝑡𝑓(𝑥)+(1−𝑡)𝑊𝑥. (108) Then 𝑥_𝑡 solves the fixed point equation 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑊𝑥_𝑡. Thus we have 𝑥_𝑡−𝑥_𝑛=(1−𝑡)𝑊𝑥_𝑡−𝑥_𝑛+𝑡𝑓𝑥_𝑡−𝑥_𝑛. (109) By Lemma 10, we conclude that 𝑥_𝑡−𝑥_𝑛²=(1−𝑡)𝑊𝑥_𝑡−𝑥_𝑛+𝑡𝑓𝑥_𝑡−𝑥_𝑛²≤(1−𝑡)²𝑊𝑥_𝑡−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑊𝑥_𝑡−𝑊𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑥_𝑡−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=(1−𝑡)²𝑥_𝑡−𝑥_𝑛²+2𝑥_𝑡−𝑥_𝑛𝑊𝑥_𝑛−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=1−2𝑡+𝑡²𝑥_𝑡−𝑥_𝑛²+𝑓_𝑛(𝑡)+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛², (110) where 𝑓_𝑛(𝑡)=(1−𝑡)²2𝑥_𝑡−𝑥_𝑛+𝑥_𝑛−𝑊𝑥_𝑛×𝑥_𝑛−𝑊𝑥_𝑛⟶0,as𝑛⟶∞. (111) It follows from (110) that 𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑥_𝑡−𝑥_𝑛²+1𝑓2𝑡^𝑛(𝑡). (112) Letting 𝑛→∞ in (112) and noticing (111), we derive limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑀₂, (113) where 𝑀₂>0 is a constant such that ‖𝑥_𝑡−𝑥_𝑛‖²≤𝑀₂ for all 𝑡∈(0,1) and 𝑛≥0. Taking 𝑡→0 in (113), we have limsup_𝑡→0limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤0. (114) On the other hand, we have 𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑓𝑞𝑥−𝑞−𝑞,𝐽_𝑛−𝑥_𝑡+𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑥_𝑡−𝑓𝑞−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑞−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡−𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡=𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑞,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑞−𝑓𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (115) It follows that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑞limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+𝜌𝑞−𝑥_𝑡limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+limsup_𝑛→∞𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (116) Taking into account that 𝑥_𝑡→𝑞 as 𝑡→0, we have limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞=limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤limsup_𝑡→0limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛𝑥−𝑞−𝐽_𝑛−𝑥_𝑡. (117) Since 𝑋 has a uniformly Gateaux differentiable norm, the duality mapping 𝐽 is norm-to-weak* uniformly continuous on bounded subsets of 𝑋. Consequently, the two limits are interchangeable and hence (107) holds. From (82), we get (𝑦_𝑛−𝑞)−(𝑥_𝑛−𝑞)→0. Noticing the norm-to-weak* uniform continuity of 𝐽 on bounded subsets of 𝑋, we deduce from (107) that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛+𝑓𝑞𝑦−𝑞−𝑞,𝐽_𝑛−𝑞−𝐽𝑥_𝑛−𝑞=limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (118)
Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. Indeed, observe that 𝑦_𝑛−𝑞²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑞+𝛼_𝑛𝑓𝑞−𝑞²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝑆_𝑛𝑦_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝜌𝑥_𝑛−𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑦_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛=−𝑞1−𝛿_𝑛−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+𝛿_𝑛𝑦_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞, (119) which implies that 𝑦_𝑛−𝑞²≤𝛼1−^𝑛1−𝜌1−𝛿_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛1−𝛿_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞. (120) Thus, we have 𝑥_𝑛+1−𝑞²≤𝜎_𝑛𝑦_𝑛−𝑞²+1−𝜎_𝑛𝐺𝑦_𝑛−𝑞²≤𝑦_𝑛−𝑞²≤𝛼1−^𝑛1−𝜌1−𝛿_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛1−𝛿_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛≤𝛼−𝑞1−^𝑛1−𝜌1−𝛿_𝑛𝑥_𝑛−𝑞²+𝛼_𝑛1−𝜌1−𝛿_𝑛2𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞.1−𝜌 (121) Applying Lemma 9 to (121), we conclude from condition (i) and (118) that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

The following results can be obtained from Theorem 23. We, therefore, omit the proof.

Corollary 24. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝐴⊂𝑋×𝑋 be an accretive operator in 𝑋 such that 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴). Let 𝐵_𝑖∶𝐶→𝑋 be 𝜁_𝑖 strictly pseudocontractive and 𝜃_𝑖 strongly accretive with 𝜃_𝑖+𝜁_𝑖≥1 for each 𝑖=1,2. Define the mapping 𝐺∶𝐶→𝐶 by 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) where 1−(𝜁_𝑖/(1+𝜁_𝑖))(1−(1−𝜃_𝑖)/𝜁_𝑖)≤𝜇_𝑖≤1 for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹=Fix(𝑆)∩Ω∩𝐴⁻¹0≠∅ where Ω=Fix(𝐺). For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆𝑦_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛Π_𝐶𝐼−𝜇₁𝐵₁×Π_𝐶𝐼−𝜇₂𝐵₂𝑦_𝑛,∀𝑛≥0. (122) Suppose that {𝑟_𝑛}⊂(0,∞), {𝜎_𝑛},{𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1], 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, and the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)∑^∞_𝑛=1(|𝜎_𝑛−𝜎_𝑛−1|+|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)<∞;(iv)∑^∞_𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞ and 𝑟_𝑛≥𝜀>0, for all 𝑛≥0 for some 𝜀>0;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1.
Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (123)

Corollary 25. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝐴⊂𝑋×𝑋 be an accretive operator in 𝑋 such that 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴). Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let 𝑉∶𝐶→𝐶 be a self-mapping such that 𝐼−𝑉∶𝐶→𝑋 is 𝜁 strictly pseudocontractive and 𝜃 strongly accretive with 𝜃+𝜁≥1. Let {𝑆_𝑖}^∞_𝑖=0 be an infinite family of nonexpansive mappings of 𝐶 into itself such that 𝐹=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Fix(𝑉)∩𝐴⁻¹0≠∅. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑦_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛×((1−𝛼)𝐼+𝛼𝑉)𝑦_𝑛,∀𝑛≥0, (124) where 1−(𝜁/(1+𝜁))(1−(1−𝜃)/𝜁)≤𝛼≤1. Suppose that {𝑟_𝑛}⊂(0,∞), {𝜎_𝑛},{𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1], 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, and the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)∑^∞_𝑛=1(|𝜎_𝑛−𝜎_𝑛−1|+|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)<∞;(iv)∑^∞_𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞ and 𝑟_𝑛≥𝜀>0, for all 𝑛≥0 for some 𝜀>0;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1.
Assume that ∑^∞_𝑛=0sup_𝑥∈𝐷‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖<∞ for any bounded subset 𝐷 of 𝐶 and let 𝑆 be a mapping of 𝐶 into itself defined by 𝑆𝑥=lim_𝑛→∞𝑆_𝑛𝑥 for all 𝑥∈𝐶 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (125)

Proof. In Theorem 23, we put 𝐵₁=𝐼−𝑉,𝐵₂=0, and 𝜇₁=𝛼 where 1−(𝜁/(1+𝜁))(1−(1−𝜃)/𝜁)≤𝛼≤1. Then GSVI (14) is equivalent to the VIP of finding 𝑥^∗∈𝐶 such that 𝐵₁𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (126) In this case, 𝐵₁∶𝐶→𝑋 is 𝜁 strictly pseudocontractive and 𝜃 strongly accretive. It is not hard to see that Fix(𝑉)=VI(𝐶,𝐵₁). As a matter of fact, we have, for 𝛼>0, 𝑢∈VI𝐶,𝐵₁⟺𝐵₁⟺𝑢,𝐽𝑦−𝑢≥0,∀𝑦∈𝐶𝑢−𝛼𝐵₁𝑢−𝑢,𝐽𝑢−𝑦≥0,∀𝑦∈𝐶⟺𝑢=Π_𝐶𝑢−𝛼𝐵₁𝑢⟺𝑢=Π_𝐶()⟺⟺(𝑉).𝑢−𝛼𝑢+𝛼𝑉𝑢𝑢−𝛼𝑢+𝛼𝑉𝑢−𝑢,𝐽𝑢−𝑦≥0,∀𝑦∈𝐶𝑢−𝑉𝑢,𝐽𝑢−𝑦≤0,∀𝑦∈𝐶⟺𝑢=𝑉𝑢⟺𝑢∈Fix (127) Accordingly, we know that 𝐹=⋂^∞_𝑖=1Fix(𝑆_𝑖)∩Ω∩𝐴⁻¹0=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Fix(𝑉)∩𝐴⁻¹0, and Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑦_𝑛=Π_𝐶𝐼−𝜇₁𝐵₁𝑦_𝑛=Π_𝐶(1−𝛼)𝑦_𝑛+𝛼𝑉𝑦_𝑛=(1−𝛼)𝑦_𝑛+𝛼𝑉𝑦_𝑛. (128) So, the scheme (53) reduces to (124). Therefore, the desired result follows from Theorem 23.

Corollary 26. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝐴⊂𝑋×𝑋 be an accretive operator in 𝑋 such that 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴). Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let 𝑉∶𝐶→𝐶 be a self-mapping such that 𝐼−𝑉∶𝐶→𝑋 is 𝜁 strictly pseudocontractive and 𝜃 strongly accretive with 𝜃+𝜁≥1. Let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹=Fix(𝑆)∩Fix(𝑉)∩𝐴⁻¹0≠∅. For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆𝑦_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛((1−𝛼)𝐼+𝛼𝑉)𝑦_𝑛,∀𝑛≥0, (129) where 1−(𝜁/(1+𝜁))(1−(1−𝜃)/𝜁)≤𝛼≤1. Suppose that {𝑟_𝑛}⊂(0,∞), {𝜎_𝑛},{𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1], 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, and the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)∑^∞_𝑛=1(|𝜎_𝑛−𝜎_𝑛−1|+|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)<∞;(iv)∑^∞_𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞ and 𝑟_𝑛≥𝜀>0, for all 𝑛≥0 for some 𝜀>0;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1.
Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (130)

Remark 27. Our Theorem 23 improves, extends, supplements, and develops Cai and Bu [10, Theorem 3.1 and Corollary 3.2] and Jung [5, Theorems 3.1] in the following aspects.
(i) The problem of finding a point 𝑞∈⋂_𝑛Fix(𝑆_𝑛)∩Ω∩𝐴⁻¹0 in our Theorem 23 is more general and more subtle than every one of both the problem of finding a point 𝑞∈⋂_𝑛Fix(𝑆_𝑛)∩Ω in Cai and Bu [10, Theorem 3.1] and the problem of finding a point 𝑞∈𝐴⁻¹0 in Jung [5, Theorem 3.1].
(ii) Our Theorem 23 drops the assumption of the asymptotical regularity of {𝑥_𝑛} in [5, Theorems 3.1] (i.e., lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0).
(iii) Cai and Bu's proof in [10, Theorem 3.1] depends on the argument techniques in [13], inequality (21) in 2-uniformly smooth Banach spaces, and inequality (27) in smooth and uniform convex Banach spaces. Jung's proof in [5, Theorem 3.1] depends on the resolvent identity in Proposition 1. It is worth emphasizing that the proof of our Theorem 23 does not depend on the argument techniques in [13], inequality (21) in 2-uniformly smooth Banach spaces, and inequality (27) in smooth and uniform convex Banach spaces. However, it depends on the resolvent identity in Proposition 1 and the inequalities in uniform convex Banach spaces; see Lemmas 13 and 17 in Section 2 of this paper.
(iv) The assumption of the uniformly convex and 2-uniformly smooth Banach space 𝑋 in [10, Theorem 3.1] is weakened to the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 23.
(v) The iterative scheme in our Theorem 23 is very different from every one in both [10, Theorem 3.1] and [5, Theorem 3.1] because the first iteration step in our iterative scheme is implicit.
(vi) The problem of finding a point 𝑞∈Fix(𝑆)∩Ω in [10, Corollary 3.2] is extended to develop the problem of finding a point 𝑞∈Fix(𝑆)∩Ω∩𝐴⁻¹0 in our Corollary 24.

4. Hybrid Explicit Extragradient Algorithm

In this section, let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. We suggest and analyze a hybrid explicit extragradient algorithm for finding a zero of 𝐴⊂𝑋×𝑋 an accretive operator in 𝑋 with 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴) and solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive self-mappings in 𝑋.

Theorem 28. Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝑋 which has a uniformly Gateaux differentiable norm. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝐴⊂𝑋×𝑋 be an accretive operator in 𝑋 such that 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴). Let 𝐵_𝑖∶𝐶→𝑋 be 𝜁_𝑖 strictly pseudocontractive and 𝜃_𝑖 strongly accretive with 𝜃_𝑖+𝜁_𝑖≥1 for each 𝑖=1,2. Define the mapping 𝐺∶𝐶→𝐶 by 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) where 1−(𝜁_𝑖/(1+𝜁_𝑖))(1−(1−𝜃_𝑖)/𝜁_𝑖)≤𝜇_𝑖≤1 for 𝑖=1,2. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Let {𝑆_𝑖}^∞_𝑖=0 be an infinite family of nonexpansive mappings of 𝐶 into itself such that 𝐹=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Ω∩𝐴⁻¹0≠∅ where Ω=Fix(𝐺). For arbitrarily given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛Π_𝐶𝐼−𝜇₁𝐵₁×Π_𝐶𝐼−𝜇₂𝐵₂𝑦_𝑛,∀𝑛≥0. (131) Suppose that {𝑟_𝑛}⊂(0,∞), {𝜎_𝑛},{𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛},{𝛿_𝑛}⊂[0,1], 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛=1, and the following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii){𝛾_𝑛},{𝛿_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1);(iii)∑^∞_𝑛=1(|𝜎_𝑛−𝜎_𝑛−1|+|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|+|𝛾_𝑛−𝛾_𝑛−1|+|𝛿_𝑛−𝛿_𝑛−1|)<∞;(iv)∑^∞_𝑛=1|𝑟_𝑛−𝑟_𝑛−1|<∞ and 𝑟_𝑛≥𝜀>0, for all 𝑛≥0 for some 𝜀>0;(v)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and 0<liminf_𝑛→∞𝜎_𝑛≤limsup_𝑛→∞𝜎_𝑛<1. Assume that ∑^∞_𝑛=0sup_𝑥∈𝐷‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖<∞ for any bounded subset 𝐷 of 𝐶 and let 𝑆 be a mapping of 𝐶 into itself defined by 𝑆𝑥=lim_𝑛→∞𝑆_𝑛𝑥 for all 𝑥∈𝐶 and suppose that Fix(𝑆)=⋂^∞_𝑖=0Fix(𝑆_𝑖). Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹, which solves the following VIP: ⟨𝑞−𝑓𝑞,𝐽𝑞−𝑝⟩≤0,∀𝑝∈𝐹. (132)

Proof. It is easy to see that (131) can be rewritten as follows 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛𝐺𝑦_𝑛,∀𝑛≥0, (133) where 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂). By Lemma 20 we know that 𝐺 is a nonexpansive self-mapping on 𝐶.
Now, let us show that the sequence {𝑥_𝑛} is bounded. Indeed, take a fixed 𝑝∈𝐹 arbitrarily. Then from (133), we have 𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝑓𝑝−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝜌𝑥_𝑛+𝑓𝑝+−𝑝−𝑝1−𝛼_𝑛𝑥_𝑛=𝛼−𝑝1−1−𝜌_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝=𝛼−𝑝1−1−𝜌_𝑛𝑥_𝑛+𝛼−𝑝1−𝜌_𝑛𝑓𝑝−𝑝,1−𝜌 (134) and hence 𝑥_𝑛+1−𝑝≤𝜎_𝑛𝑦_𝑛+−𝑝1−𝜎_𝑛𝐺𝑦_𝑛≤𝑦−𝑝_𝑛≤𝛼−𝑝1−1−𝜌_𝑛𝑥_𝑛+𝛼−𝑝1−𝜌_𝑛𝑓𝑝−𝑝𝑥1−𝜌≤max_𝑛,𝑓𝑝−𝑝−𝑝.1−𝜌 (135) By induction, we obtain 𝑥_𝑛−𝑝≤max𝑥₀,𝑓𝑝−𝑝−𝑝1−𝜌,∀𝑛≥0. (136) It immediately follows that {𝑥_𝑛} is bounded and so are {𝑦_𝑛}, {𝐺𝑦_𝑛}, {𝑓(𝑥_𝑛)}, {𝐽_𝑟𝑛𝑥_𝑛}, and {𝑆_𝑛𝑥_𝑛}.
Let us show that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (137) As a matter of fact, observe that 𝑦_𝑛 can be rewritten as follows: 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛, (138) where 𝑧_𝑛=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛)/(1−𝛽_𝑛). Observe that 𝑧_𝑛−𝑧_𝑛−1=𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛1−𝛽_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛+𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1≤𝑦_𝑛−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛+𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−11−𝛽_𝑛−1=11−𝛽_𝑛𝑦_𝑛−𝛽_𝑛𝑥_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛−1𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛𝑦_𝑛−𝛽_𝑛𝑥_𝑛−𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1=11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1−𝛿_𝑛−1𝑆_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑆_𝑛−1𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1. (139) On the other hand, repeating the same arguments as those of (66) in the proof of Theorem 23, we can derive 𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+𝑀₀𝑟_𝑛−1−𝑟_𝑛,∀𝑛≥1, (140) where sup_𝑛≥1{(1/𝜀)(‖𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1‖+‖𝐽_𝑟𝑛−1𝑥_𝑛−1−𝑥_𝑛‖)}≤𝑀₀ for some 𝑀₀>0. Substituting (140) into (139), we have 𝑧_𝑛−𝑧_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛾_𝑛𝑥_𝑛−1−𝑥_𝑛+𝑀₀𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑆_𝑛−1𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛𝑦_𝑛−1−𝛽_𝑛−1𝑥_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛𝑥_𝑛−1−𝑥_𝑛+𝑀₀𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑆_𝑛−1𝑥_𝑛−1+𝛿_𝑛𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−11−𝛽_𝑛−11−𝛽_𝑛×𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛾_𝑛−1𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛿_𝑛−1𝑆_𝑛−1𝑥_𝑛−1≤11−𝛽_𝑛𝛼_𝑛𝜌+𝛾_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛿_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛𝛽_𝑛−𝛽_𝑛−1𝑀=11−𝛽_𝑛1−𝛼_𝑛−𝛽_𝑛+𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛𝛽_𝑛−𝛽_𝑛−1𝑀=1−1−𝜌1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀1−𝛽_𝑛𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛1−𝛽_𝑛𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛𝛽_𝑛−𝛽_𝑛−1𝑀, (141) where sup_𝑛≥0{𝑀₀+‖𝑓(𝑥_𝑛)‖+‖𝐽_𝑟𝑛𝑥_𝑛‖+‖𝑆_𝑛𝑥_𝑛‖}≤𝑀 for some 𝑀>0. Also, from (133) we have 𝑥_𝑛+1−𝑥_𝑛=𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1+1−𝜎_𝑛𝐺𝑦_𝑛−𝐺𝑦_𝑛−1. (142) Taking into account condition (v), we may assume, without loss of generality, that {𝛽_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1). This together with (141) implies that 𝑥_𝑛+1−𝑥_𝑛≤𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1+1−𝜎_𝑛𝐺𝑦_𝑛−𝐺𝑦_𝑛−1≤𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1+1−𝜎_𝑛𝑦_𝑛−𝑦_𝑛−1=𝑦_𝑛−𝑦_𝑛−1+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1≤1−1−𝜌1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀1−𝛽_𝑛𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝛿_𝑛1−𝛽_𝑛𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+11−𝛽_𝑛−11−𝛽_𝑛𝛽_𝑛−𝛽_𝑛−1𝑀+𝜎_𝑛−𝜎_𝑛−1𝑦_𝑛−1−𝐺𝑦_𝑛−1≤1−1−𝜌1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑀₁+𝜎_𝑛−𝜎_𝑛−1𝑀₁=1−1−𝜌1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝜎_𝑛−𝜎_𝑛−1+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1+𝑟_𝑛−1−𝑟_𝑛+𝑆_𝑛𝑥_𝑛−1−𝑆_𝑛−1𝑥_𝑛−1, (143) where sup_𝑛≥0{(𝑀/(1−𝑏)²)+‖𝑦_𝑛−𝐺𝑦_𝑛‖}≤𝑀₁ for some 𝑀₁>0. Since ∑^∞_𝑛=0𝛼_𝑛=∞ and ((1−𝜌)/(1−𝛽_𝑛))𝛼_𝑛≥(1−𝜌)𝛼_𝑛, we know that ∑^∞_𝑛=0((1−𝜌)/(1−𝛿_𝑛))𝛼_𝑛=∞. So, applying Lemma 9 to (143), we obtain from conditions (iii) and (iv) and the assumption on {𝑆_𝑛} that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (144)
Next we show that ‖𝑥_𝑛−𝐺𝑥_𝑛‖→0 as 𝑛→∞.
Indeed, according to Lemma 10 we have from (133) 𝑦_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝≤𝛼_𝑛𝜌𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛=𝛼−𝑝_𝑛𝜌+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛≤−𝑝1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝. (145) Utilizing Lemma 17 we get from (133) and (145) 𝑥_𝑛+1−𝑝²=𝜎_𝑛𝑦_𝑛+−𝑝1−𝜎_𝑛𝐺𝑦_𝑛−𝑝²≤𝜎_𝑛𝑦_𝑛−𝑝²+1−𝜎_𝑛𝐺𝑦_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛≤𝜎_𝑛𝑦_𝑛−𝑝²+1−𝜎_𝑛𝑦_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛=𝑦_𝑛−𝑝²−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛≤1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝,𝐽𝑦_𝑛−𝑝−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛≤𝑥_𝑛−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑦_𝑛−𝑝−𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛, (146) which hence yields 𝜎_𝑛1−𝜎_𝑛𝑔₁𝑦_𝑛−𝐺𝑦_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝛼_𝑛𝑓𝑝−𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+2𝛼_𝑛𝑓𝑝−𝑝𝑦_𝑛−𝑝. (147) Since 𝛼_𝑛→0 and ‖𝑥_𝑛+1−𝑥_𝑛‖→0, from condition (v) and the boundedness of {𝑥_𝑛} and {𝑦_𝑛}, it follows that lim_𝑛→∞𝑔₁𝑦_𝑛−𝐺𝑦_𝑛=0. (148) Utilizing the properties of 𝑔₁, we have lim_𝑛→∞𝑦_𝑛−𝐺𝑦_𝑛=0. (149) Observe that 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑦_𝑛=𝑥_𝑛−𝑥_𝑛+1+1−𝜎_𝑛𝐺𝑦_𝑛−𝑦_𝑛⟶0,as𝑛⟶∞, (150) and hence 𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐺𝑦_𝑛+𝐺𝑦_𝑛−𝐺𝑥_𝑛≤2𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐺𝑦_𝑛⟶0,as𝑛⟶∞. (151) That is, lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0,lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (152) Next, let us show that lim_𝑛→∞𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛=0,lim_𝑛→∞𝑆_𝑛𝑥_𝑛−𝑥_𝑛=0. (153) Indeed, observe that 𝑦_𝑛 can be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛+𝛿_𝑛𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛾_𝑛+𝛿_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝑒_𝑛𝑧_𝑛, (154) where 𝑒_𝑛=𝛾_𝑛+𝛿_𝑛 and 𝑧_𝑛=(𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛)/(𝛾_𝑛+𝛿_𝑛). Utilizing Lemma 13 and (154), we have 𝑦_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝑒_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑓(𝑥_𝑛)−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝑒_𝑛𝑧_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₃𝑧_𝑛−𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿−𝑝_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓(𝑥_𝑛)−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛+𝑒_𝑛𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛+𝛿−𝑝_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²=𝛼_𝑛𝑓(𝑥_𝑛)−𝑝²+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛, (155) which hence implies that 𝛽_𝑛𝑒_𝑛𝑔₂𝑧_𝑛−𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (156) Utilizing (152), conditions (i), (ii), and (v) and the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, and {𝑓(𝑥_𝑛)}, we get lim_𝑛→∞𝑔₂𝑧_𝑛−𝑥_𝑛=0. (157) From the properties of 𝑔₂, we have lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (158) Utilizing Lemma 17 and the definition of 𝑧_𝑛, we have 𝑧_𝑛−𝑝²=𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛾_𝑛+𝛿_𝑛−𝑝²=𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿−𝑝^𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₃𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝛾_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²+𝛿_𝑛𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₃𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛=𝑥_𝑛−𝑝²−𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₃𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛, (159) which leads to 𝛾_𝑛𝛿_𝑛𝛾_𝑛+𝛿_𝑛²𝑔₃𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝑥_𝑛−𝑝²−𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑧_𝑛−𝑝𝑥_𝑛−𝑧_𝑛. (160) Since {𝑥_𝑛} and {𝑧_𝑛} are bounded and ‖𝑧_𝑛−𝑥_𝑛‖→0 as 𝑛→∞, we deduce from condition (ii) that lim_𝑛→∞𝑔₃𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (161) From the properties of 𝑔₃, we have lim_𝑛→∞𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (162) On the other hand, 𝑦_𝑛 can also be rewritten as follows: 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛼_𝑛+𝛿_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛼_𝑛+𝛿_𝑛=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝑑_𝑛𝑧_𝑛, (163) where 𝑑_𝑛=𝛼_𝑛+𝛿_𝑛 and 𝑧_𝑛=(𝛼_𝑛𝑓(𝑥_𝑛)+𝛿_𝑛𝑆_𝑛𝑥_𝑛)/(𝛼_𝑛+𝛿_𝑛). Utilizing Lemma 13 and the convexity of ‖⋅‖², we have 𝑦_𝑛−𝑝²=𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝+𝑑_𝑛𝑧_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝑧_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝑓𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛𝛼_𝑛+𝛿_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛+𝛿−𝑝_𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+𝑑_𝑛𝛼_𝑛𝛼_𝑛+𝛿_𝑛𝑓𝑥_𝑛−𝑝²+𝛿_𝑛𝛼_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝛼_𝑛𝑓(𝑥_𝑛)−𝑝²+𝑥_𝑛−𝑝²−𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛, (164) which hence implies that 𝛽_𝑛𝛾_𝑛𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝛼_𝑛𝑓(𝑥_𝑛)−𝑝²+𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (165) From (152), conditions (i), (ii), and (v), and the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, and {𝑓(𝑥_𝑛)}, we have lim_𝑛→∞𝑔₄𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=0. (166) Utilizing the properties of 𝑔₄, we have lim_𝑛→∞𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=0. (167) Note that 𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛+𝐽_𝑟𝑛𝑥_𝑛−𝑆_𝑛𝑥_𝑛. (168) From (162) and (167), we get lim_𝑛→∞𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (169) In terms of (169) and Lemma 14, we have 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝑆_𝑛𝑥_𝑛−𝑆𝑥_𝑛⟶as𝑛⟶∞. (170) That is, lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (171) Furthermore, repeating the same arguments as those of (104) in the proof of Theorem 23, we can conclude that lim_𝑛→∞𝑥_𝑛−𝐽_𝑟𝑥_𝑛=0 (172) for a fixed number 𝑟 such that 𝜀>𝑟>0.
Define a mapping 𝑊𝑥=(1−𝜃₁−𝜃₂)𝐽_𝑟𝑥+𝜃₁𝑆𝑥+𝜃₂𝐺𝑥, where 𝜃₁,𝜃₂∈(0,1) are two constants with 𝜃₁+𝜃₂<1. Then by Lemma 16, we have that Fix(𝑊)=Fix(𝐽_𝑟)∩Fix(𝑆)∩Fix(𝐺)=𝐹. We observe that 𝑥_𝑛−𝑊𝑥_𝑛=1−𝜃₁−𝜃₂𝑥_𝑛−𝐽_𝑟𝑥_𝑛+𝜃₁𝑥_𝑛−𝑆𝑥_𝑛+𝜃₂𝑥_𝑛−𝐺𝑥_𝑛≤1−𝜃₁−𝜃₂𝑥_𝑛−𝐽_𝑟𝑥_𝑛+𝜃₁𝑥_𝑛−𝑆𝑥_𝑛+𝜃₂𝑥_𝑛−𝐺𝑥_𝑛. (173) From (152), (171), and (172), we obtain lim_𝑛→∞𝑥_𝑛−𝑊𝑥_𝑛=0. (174)
Repeating the same arguments as those of (107) in the proof of Theorem 23, we can deduce that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑥_𝑛−𝑞≤0. (175) Since 𝐽 is norm-to-weak* uniformly continuous over bounded subsets of 𝑋, we obtain from ‖𝑥_𝑛−𝑦_𝑛‖→0 that limsup_𝑛→∞𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤0. (176) Finally, let us show that 𝑥_𝑛→𝑞 as 𝑛→∞. Indeed, observe that 𝑦_𝑛−𝑞²=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝑓𝑞−𝑞²≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝛽_𝑛𝑥_𝑛−𝑞+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑞²+𝛿_𝑛𝑆_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞≤𝛼_𝑛𝜌𝑥_𝑛−𝑞²+𝛽_𝑛𝑥_𝑛−𝑞²+𝛾_𝑛𝑥_𝑛−𝑞+𝛿_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛=𝛼−𝑞_𝑛𝜌+𝛽_𝑛+𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛=−𝑞1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞, (177) and hence 𝑥_𝑛+1−𝑞²≤𝜎_𝑛𝑦_𝑛−𝑞²+1−𝜎_𝑛𝐺𝑦_𝑛−𝑞²≤𝑦_𝑛−𝑞²≤1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑞²+2𝛼_𝑛𝑓𝑞−𝑞,𝐽𝑦_𝑛−𝑞. (178) Applying Lemma 9 to (178), we conclude from condition (i) and (178) that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

The following results can be obtained from Theorem 28. We, therefore, omit the proof.

Corollary 29. Under the same conditions as those in Corollary 24, let {𝑥_𝑛} be the sequence generated from any given 𝑥₀∈𝐶 by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆𝑥_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛Π_𝐶𝐼−𝜇₁𝐵₁×Π_𝐶𝐼−𝜇₂𝐵₂𝑦_𝑛,∀𝑛≥0. (179) Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹=Fix(𝑆)∩Ω∩𝐴⁻¹0, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (180)

Corollary 30. Under the same conditions as those in Corollary 25, let {𝑥_𝑛} be the sequence generated from any given 𝑥₀∈𝐶 by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆_𝑛𝑥_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛((1−𝛼)𝐼+𝛼𝑉)𝑦_𝑛,∀𝑛≥0. (181) Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹=⋂^∞_𝑖=0Fix(𝑆_𝑖)∩Fix(𝑉)∩𝐴⁻¹0   , which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (182)

Corollary 31. Under the same conditions as those in Corollary 26, let {𝑥_𝑛} be the sequence generated from any given 𝑥₀∈𝐶 by 𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝐽_𝑟𝑛𝑥_𝑛+𝛿_𝑛𝑆𝑥_𝑛,𝑥_𝑛+1=𝜎_𝑛𝑦_𝑛+1−𝜎_𝑛((1−𝛼)𝐼+𝛼𝑉)𝑦_𝑛,∀𝑛≥0. (183) Then {𝑥_𝑛} converges strongly to 𝑞∈𝐹=Fix(𝑆)∩Fix(𝑉)∩𝐴⁻¹0, which solves the following VIP: 𝑞−𝑓𝑞,𝐽𝑞−𝑝≤0,∀𝑝∈𝐹. (184)

Remark 32. Our Theorem 28 improves, extends, supplements, and develops Cai and Bu [10, Theorem 3.1 and Corollary 3.2] and Jung [5, Theorems 3.1] in the following aspects.
(i) The problem of finding a point 𝑞∈⋂_𝑛Fix(𝑆_𝑛)∩Ω∩𝐴⁻¹0 in our Theorem 28 is more general and more subtle than every one of both the problem of finding a point 𝑞∈⋂_𝑛Fix(𝑆_𝑛)∩Ω in Cai and Bu [10, Theorem 3.1] and the problem of finding a point 𝑞∈𝐴⁻¹0 in Jung [5, Theorem 3.1].
(ii) Our Theorem 28 drops the assumption of the asymptotical regularity of {𝑥_𝑛} in [5, Theorems 3.1] (i.e., lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0).
(iii) Cai and Bu's proof in [10, Theorem 3.1] depends on the argument techniques in [13], inequality (21) in 2-uniformly smooth Banach spaces, and inequality (27) in smooth and uniform convex Banach spaces. Jung's proof in [5, Theorem 3.1] depends on the resolvent identity in Proposition 1. It is worth emphasizing that the proof of our Theorem 28 does not depend on the argument techniques in [13], inequality (21) in 2-uniformly smooth Banach spaces, and inequality (27) in smooth and uniform convex Banach spaces. However, it depends on the resolvent identity in Proposition 1 and the inequalities in uniform convex Banach spaces; see Lemmas 13 and 17 in Section 2 of this paper.
(iv) The assumption of the uniformly convex and 2-uniformly smooth Banach space 𝑋 in [10, Theorem 3.1] is weakened to the one of the uniformly convex Banach space 𝑋 having a uniformly Gateaux differentiable norm in our Theorem 28.
(v) The iterative scheme in our Theorem 28 is very different from every one in both [10, Theorem 3.1] and [5, Theorem 3.1] because the first iteration step in the iterative scheme of [10, Theorem 3.1] is given by 𝑦_𝑛=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂)𝑥_𝑛 and the first iteration step in the iterative scheme of [5, Theorem 3.1] is given by 𝑦_𝑛=𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)𝐽_𝑟𝑛𝑥_𝑛. In the meantime, it is clear that the second iteration steps for 𝑥_𝑛+1 in three iterative schemes are completely different.
(vi) The problem of finding a point 𝑞∈Fix(𝑆)∩Ω in [10, Corollary 3.2] is extended to develop the problem of finding a point 𝑞∈Fix(𝑆)∩Ω∩𝐴⁻¹0 in our Corollary 29.

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This research was partially supported by a Grant from NSC 102-2115-M-037-001.